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Question: Statement: 1 remainder when \({{99}^{100}}\) is divided by 10 is 2 because Statement: 2\(^{n}{{C}_...

Statement: 1 remainder when 99100{{99}^{100}} is divided by 10 is 2 because
Statement: 2nCr=nCn−r^{n}{{C}_{r}}{{=}^{n}}{{C}_{n-r}}
A.Statement 1 is true, statement 2 is true
Statement 2 is correct explanation for statement 1
B.Statement 1 is true statement 2 is true, statement2 is not a correct explanation for statement 1
C.Statement 1 is true, statement 2 is false
D.Statement 1 is false, statement 1 is true

Explanation

Solution

1st is to check even & odd power of 9 at unit place & 2nd statement is the rule for combination that if nCx=xCy^{n}{{C}_{x}}{{=}^{x}}{{C}_{y}} then either x=y or x+y=nx+y=n

Formula used:
1st Statement we have 91=9&92=81{{9}^{1}}=9\And {{9}^{2}}=81
9 at unit place when power is odd & 1 at unit place if power of 9 is even.
In 2nd Statement we havenCr=nCy^{n}{{C}_{r}}{{=}^{n}}{{C}_{y}} then x=y or x+y=nx+y=n
nCr=n!(n−r)!r! 0≤r≤n^{n}{{C}_{r}}=\frac{n!}{\left( n-r \right)!r!}\text{ 0}\le \text{r}\le \text{n}

Complete step-by-step answer:
99100{{99}^{100}} we check the unit digit
For
91=9 92=81 \begin{aligned} & {{9}^{1}}=9 \\\ & {{9}^{2}}=81 \\\ \end{aligned}
1 at unit place
9 at unit place
93=9{{9}^{3}}=9 at unit place 94=1{{9}^{4}}=1 at unit place
So for away odd powers of 9 we have 1 unit. So first statement is false
Statement2: nCr=nCn−r^{n}{{C}_{r}}{{=}^{n}}{{C}_{n-r}}
LHS
nCr=n!(n−r)!n! 0=r≤n^{n}{{C}_{r}}=\frac{n!}{\left( n-r \right)!n!}\text{ 0=}r\le n (general formula)
RHS
nCr=n![n−(n−r)]!(n−r)! =n!(n−n+r)!(n−r)! =n!r!(n−r)!=n!(n−r)!r!=nCr=LHS \begin{aligned} & ^{n}{{C}_{r}}=\frac{n!}{\left[ n-\left( n-r \right) \right]!\left( n-r \right)!} \\\ & =\frac{n!}{\left( n-n+r \right)!\left( n-r \right)!} \\\ & =\frac{n!}{r!\left( n-r \right)!}=\frac{n!}{\left( n-r \right)!r!}{{=}^{n}}{{C}_{r}}=LHS \\\ \end{aligned}
Hence statement 2 is true.
Also
Statement 2 is not a correct explanation for statement 1
Answer option is (D)
Additional information:
Statement 1 is from the topic number theory that deals with power of unit digit in the number while statement 2 is called deals with combination. Whereas this particular statement 2 is a rule for combination.

Note: In this question, we should firstly check if the statements are correct or not then we find that is there any relation between them. Finally we see which of the four options best suits as.